Ising Quantum Chain and Sequence Evolution

A sequence space model which describes the interplay of mutation and selection in molecular evolution is shown to be equivalent to an Ising quantum chain. Observable quantities tailored to match the biological situation are then employed to treat three fitness landscapes exactly.

[1]  Chen Ning Yang,et al.  The Spontaneous Magnetization of a Two-Dimensional Ising Model , 1952 .

[2]  R. B. Potts,et al.  The Combinatrial Method and the Two-Dimensional Ising Model , 1955 .

[3]  E. Lieb,et al.  Two Soluble Models of an Antiferromagnetic Chain , 1961 .

[4]  Renfrey B. Potts,et al.  Correlations and Spontaneous Magnetization of the Two‐Dimensional Ising Model , 1963 .

[5]  Elliott H. Lieb,et al.  Two-Dimensional Ising Model as a Soluble Problem of Many Fermions , 1964 .

[6]  Irene A. Stegun,et al.  Handbook of Mathematical Functions. , 1966 .

[7]  Large groups of automorphisms of C*-algebras , 1967 .

[8]  E. Størmer.  Symmetric states of infinite tensor products of C*-algebras , 1969 .

[9]  M. Kimura,et al.  An introduction to population genetics theory , 1971 .

[10]  P. Pfeuty.  The one-dimensional Ising model with a transverse field , 1970 .

[11]  Colin J. Thompson,et al.  Mathematical Statistical Mechanics , 1972 .

[12]  Colin J. Thompson,et al.  On Eigen's theory of the self-organization of matter and the evolution of biological macromolecules , 1974 .

[13]  O. Bratteli.  Operator Algebras And Quantum Statistical Mechanics , 1979 .

[14]  W. Ewens.  Mathematical Population Genetics , 1980 .

[15]  M. Fannes,et al.  Equilibrium states for mean field models , 1980 .

[16]  J. Hofbauer.  The selection mutation equation , 1985, Journal of mathematical biology.

[17]  Silviu Guiasu,et al.  The principle of maximum entropy , 1985 .

[18]  N. Barton.  The maintenance of polygenic variation through a balance between mutation and stabilizing selection. , 1986, Genetical research.

[19]  R. Bürger.  On the maintenance of genetic variation: global analysis of Kimura's continuum-of-alleles model , 1986, Journal of mathematical biology.

[20]  L. P. Kok,et al.  Table erratum: A table of series and products [Prentice-Hall, Englewood Cliffs, N.J., 1975] by E. R. Hansen , 1986 .

[21]  I. Leuthäusser,et al.  An exact correspondence between Eigen’s evolution model and a two‐dimensional Ising system , 1986 .

[22]  I. Leuthäusser,et al.  Statistical mechanics of Eigen's evolution model , 1987 .

[23]  D S Rumschitzki.  Spectral properties of Eigen evolution matrices , 1987, Journal of mathematical biology.

[24]  M. Eigen,et al.  Molecular quasi-species. , 1988 .

[25]  R. Werner,et al.  Quantum Statistical Mechanics of General Mean Field Systems , 1989 .

[26]  M. Schlottmann,et al.  The Ising quantum chain with defects (I). The exact solution , 1989 .

[27]  M. Baake,et al.  The Ising quantum chain with defects (II). The so(2n)Kac-Moody Spectra , 1989 .

[28]  The Quantum Statistical Free Energy Minimum Principle for Multi-Lattice Mean Field Theories , 1990 .

[29]  J. Gillespie.  The causes of molecular evolution , 1991 .

[30]  N. Barton,et al.  Natural and sexual selection on many loci. , 1991, Genetics.

[31]  S. Kauffman,et al.  Coevolution to the edge of chaos: coupled fitness landscapes, poised states, and coevolutionary avalanches. , 1991, Journal of theoretical biology.

[32]  Tarazona.  Error thresholds for molecular quasispecies as phase transitions: From simple landscapes to spin-glass models. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[33]  H. Frahm.  Integrable spin-1/2 XXZ Heisenberg chain with competing interactions , 1992 .

[34]  Internal symmetries and limiting Gibbs states in quantum lattice mean-field theories , 1993 .

[35]  L. Peliti,et al.  An evolutionary version of the random energy model , 1993 .

[36]  Diploid models on sequence space , 1995 .

[37]  P. Schuster,et al.  Error propagation in reproduction of diploid organisms. A case study on single peaked landscapes. , 1995, Journal of theoretical biology.

[38]  P. Higgs,et al.  Population evolution on a multiplicative single-peak fitness landscape. , 1996, Journal of theoretical biology.

[39]  S. Galluccio,et al.  Diffusion on a hypercubic lattice with pinning potential: exact results for the error-catastrophe problem in biological evolution , 1996, cond-mat/9601088.

[40]  Yi-Cheng Zhang.  QUASISPECIES EVOLUTION OF FINITE POPULATIONS , 1997 .

[41]  T. Wiehe,et al.  Bifurcations in haploid and diploid sequence space models , 1997 .

[42]  M. Baake,et al.  Ising quantum chain is equivalent to a model of biological evolution , 1997 .

[43]  Thermodynamic Formalism and Phase Transitions of Generalized Mean-Field Quantum Lattice Models , 1998 .

[44]  Michael Baake,et al.  Quantum mechanics versus classical probability in biological evolution , 1998 .